I found that the Ring skill for Alexa does not yet support arming the alarm. Disappointing that it allows it via voice command but not through a routine.
Based on other forums I have found a workaround until the Ring skill for Alexa is expanded. I used the app SimpleCommands. It has a skill for Ring that does support the alarm base station. And there is a skill called Kloee for SC for Alexa that allows you to use that capability in an Alexa routine. I created a routine in SimpleCommands to arm the alarm. The SimpleCommands routine shows up in Alexa routines (Add action, Smart Home, Control scene, name of SimpleCommand routine).
I can now say "Alexa, Goodbye" and the lights are turned off and the alarm is armed in away mode.